The distribution of all possible 5-card poker hands is well known. With a standard deck of 52 cards, there are C(52,5), or 2,598,960 ways to select five distinct cards (where order does not matter). If we determine the hand value for each of those approximately 2.6 million poker hands, we get the following table:

Five Card Poker Hands | |
---|---|

40 | Straight Flush |

624 | Four of a Kind |

3,744 | Full House |

5,108 | Flush |

10,200 | Straight |

54,912 | Three of a Kind |

123,552 | Two Pair |

1,098,240 | One Pair |

1,302,540 | High Card |

2,598,960 | TOTAL |

It is this table that initially gave us our poker hand rankings. Notice that the fewer the ways available to make a particular poker hand, the higher its value. That's why a Straight Flush beats a Four-of-a-Kind. There are only 40 ways to make a Straight Flush, but 624 ways to make a Four-of-a-Kind. I have sometimes seen the above table given where a Royal Flush is listed separately, and ranked directly above a Straight Flush. In my opinion, this is unnecessary, since a Royal Flush is simply a special case of a Straight Flush (it just happens to be the highest Straight Flush possible). We don't single out Four Aces with a King kicker as being the highest possible Four-of-a-Kind hand, so there is no need to single out the Royal Flush hand either. But some tables list it that way, so just be aware that I've have combined the Royals with the other Straight Flushes.

Now, recently I got to thinking about wild card games, and I wondered what the frequency table looked like when you counted Deuces as being wild. Since the rules for wild card games are numerous and definately not standardized, I used the following stipulations:

- Five-of-a-Kind hands are allowed.
- Wild cards can represent any card (no restrictions).
- If a hand contains one or more wild cards, that hand's value is the highest possible it can be.
- Double-Ace and Triple-Ace Flushes are not allowed.

- Four of a Kind (KKKKJ)
- Full House (KKKJJ or JJJKK)
- Three of a Kind (KKKJx)
- Two Pair (KKJJx, AAKKJ, etc.)
- One Pair (KKJxx)

My first thought was to search the internet and check poker books to
see if such a table already existed. The internet turned up nothing,
but the book "Scarne's Guide to Modern Poker" had the table I was looking for.
However, I never take such tables as gospel, and I wanted to compute
and verify the values myself (which is a good thing, as I soon learned
that Scarne's table was incorrect!) I also posted this question in
the *rec.gambling.poker* newsgroup, and a number of RPG'ers posted
results they had either found or computed. Even then, some of the
figures given were incorrect or inconsistant; so I decided to determine
the hand frequencies myself.

There were a couple of ways I could tackle this problem. One way would be
to use math theory and combinatorics to enumerate all the hand rank frequencies.
Another would be to write some poker code to loop over all 2,598,960 possible
five card poker hands, and determine each hand's value if Deuces were wild.
I decided to do both, since I *should* get the same results using either method,
and it would act as a sanity check.

I had already written some 'C' code that determined the value of a specific five card poker hand. It required just a little massaging to get it to work with Deuces being wild. After lots of debugging and code checking, I finally obtained the following results:

Poker Hand Frequencieswith Deuces Wild | |

672 | Five of a Kind |

2,552 | Straight Flush |

31,552 | Four of a Kind |

12,672 | Full House |

14,472 | Flush |

62,232 | Straight |

355,080 | Three of a Kind |

95,040 | Two Pair |

1,225,008 | One Pair |

799,680 | High Card |

2,598,960 | TOTAL |

Now the question was whether or not these figures were correct. Although I was fairly certain my code was correct, I'd been burned before by making invalid assumptions in my coding logic. This meant to be absolutely sure, I needed to validate my results by using math and combinatorics.

Before we start, a quick note about combinatorics. There are often
multiple ways to obtain the same answer. In the tables given below,
I will give the combinatoric method I used to obtain my answer. You,
however, may come up with a completely different way to compute the same
answer. That is okay. It all depends on how you "count" your objects.
For example, let's say we remove all four Deuces from the deck, and we
wish to figure out how many ways we can select five cards of differing
ranks. One person might look at the problem this way:
There are 48 cards to choose from for our first card. Let's say we select the King of Spades. For our next card, we only have 44 to choose from (because we can't select another King). We select the Six of Diamonds. For the third card, we only have 40 to choose from (because we can't select any Kings or Sixes), and so on. So selecting all five cards gives usNow, a second person might come along and reason as thus: Okay, there are twelve ranks to choose from since the Deuces have been removed. I need to choose five ranks out of the possible twelve, so that's C(12,5), or 792 combinations. Now that I have chosen my five distinct ranks, each of those five cards can be one of four possible suits. So I need to multiply 792 by 4See? Two different ways to tackle the problem, but they both come up with the same result. |

Okay, enough talk. Let's start enumerating. First off, I determined how many five card poker hands contained zero Deuces, one Deuce, two Deuces, three Deuces, or all four Deuces. Combinatorics made this very easy, and the following table shows the results:

Breakdown of Deuce Counts | ||
---|---|---|

48 | hands with four Deuces | C(4,4) x C(48,1) |

4,512 | hands with three Deuces | C(4,3) x C(48,2) |

103,776 | hands with two Deuces | C(4,2) x C(48,3) |

778,320 | hands with one Deuce | C(4,1) x C(48,4) |

1,712,304 | hands with zero Deuces | C(4,0) x C(48,5) |

2,598,960 |
TOTAL |

Next, I tackled each of the five possibilities shown above. The first case was easy. If you have all four Deuces in your five card poker hand, then you obviously have a Five-of-a-Kind. So out of the 48 possible hands that contain four Deuces, all of them are Five-of-a-Kinds.

Hands with Four Deuces | ||
---|---|---|

48 | Five of a Kind | C(48,1) |

48 | TOTAL |

Next, I took a look at the hands with three Deuces. After some thought,
you should realize that if you hold three Deuces, then the
absolute ** minimum** hand you could hold is Four-of-a-Kind.
If the other two non-wild cards happen to be a pair, then you have a
Five-of-a-Kind. If they are the same suit and close enough in rank, you
have a Straight Flush. Otherwise, you must have a Four-of-a-Kind.
No other hands are possible with three wild Deuces.
Since there are four Deuces in a deck, but we only need to choose three of
them, each of the counts will have a factor of C(4,3) for the various
combinations of three Deuces. For Five-of-a-Kind hands, there are 12
ranks to select from (Ace through Trey) for the pair, and C(4,2) ways
to form that pair once the rank has been selected. That gives us
288 Five-of-a-Kind hands. Straight Flush hands
are more tricky to compute. We must enumerate all ways to form a
Straight Flush using two non-wild cards. It turns out there are 41 such
ways, as shown in the table below. Note that when Deuces are wild,
Six-High Straight Flushes are verboten since one can always make a higher
Straight Flush.

Straight Flushes with Three Deuces (41 ways) |
|||||
---|---|---|---|---|---|

Royal Flush | AK222 | AQ222 | AJ222 | AT222 | KQ222 |

KJ222 | KT222 | QJ222 | QT222 | JT222 | |

King-High Straight Flush | K9222 | Q9222 | J9222 | T9222 | |

Queen-High Straight Flush | Q8222 | J8222 | T8222 | 98222 | |

Jack-High Straight Flush | J7222 | T7222 | 97222 | 87222 | |

Ten-High Straight Flush | T6222 | 96222 | 86222 | 76222 | |

Nine-High Straight Flush | 95222 | 85222 | 75222 | 65222 | |

Eight-High Straight Flush | 84222 | 74222 | 64222 | 54222 | |

Seven-High Straight Flush | 73222 | 63222 | 53222 | 43222 | |

Five-High Straight Flush | 5A222 | 4A222 | 3A222 |

With that table, I can now calculate how many Three Deuce Straight Flush hands there are. There are C(4,3) ways to choose three of the four Deuces, times 41 ways to make a Straight Flush, times 4 different suits, giving a total of 656. Once I've eliminated all Five-of-a-Kind and Straight Flush hands, only Four-of-a-Kind are left remaining. I now know the totals for hands with three Deuces.

Hands with Three Deuces |
||
---|---|---|

288 | Five of a Kind | C(4,3) x 12 x C(4,2) |

656 | Straight Flush | C(4,3) x 41 x 4 |

3,568 | Four of a Kind | 4,512 - 288 - 656 |

4,512 |
TOTAL |

Next, I worked on the hands containing two Deuces. Note that when holding two wild Deuces, the minimum hand possible is Three-of-a-Kind. Also observe that it is impossible to have a Full House when holding two wild cards. The combinatorics used to calculate the hand frequencies are similar to how we computed them for the three Deuce hands. It turns out there are 55 ways to make a Straight Flush this time (rather than 41 as before). For Flush hands, we select any three cards (except the Deuce) of the same suit, but then subtract off the Straight Flush hands since those would be included. For Straights, we use the same 55 ways to make Straight Flushes, but this time we make sure the suits don't all match. Three-of-a-Kinds would be a real bear to compute, so we take the easy way out. Once we've computed all the other hand frequencies, anything left remaining has to be a Three-of-a-Kind.

Straight Flushes with Two Deuces (55 ways) |
||||||
---|---|---|---|---|---|---|

Royal Flush | AKQ22 | AKJ22 | AKT22 | AQJ22 | AQT22 | |

AJT22 | KQJ22 | KQT22 | KJT22 | QJT22 | ||

King-High Straight Flush | KQ922 | KJ922 | KT922 | QJ922 | QT922 | JT922 |

Queen-High Straight Flush | QJ822 | QT822 | Q9822 | JT822 | J9822 | T9822 |

Jack-High Straight Flush | JT722 | J9722 | J8722 | T9722 | T8722 | 98722 |

Ten-High Straight Flush | T9622 | T8622 | T7622 | 98622 | 97622 | 87622 |

Nine-High Straight Flush | 98522 | 97522 | 96522 | 87522 | 86522 | 76522 |

Eight-High Straight Flush | 87422 | 86422 | 85422 | 76422 | 75422 | 65422 |

Seven-High Straight Flush | 76322 | 75322 | 74322 | 65322 | 64322 | 54322 |

Five-High Straight Flush | 54A22 | 53A22 | 43A22 |

Hands with Two Deuces | ||
---|---|---|

288 | Five of a Kind | C(4,2) x 12 x C(4,3) |

1,320 | Straight Flush | C(4,2) x 55 x 4 |

19,008 | Four of a Kind | C(4,2) x C(4,2) x 12 x 44 |

0 | Full House | not possible |

3,960 | Flush | C(4,2) x C(12,3) x 4 - 1,320 |

19,800 | Straight | C(4,2) x 55 x 4^{3} - 1,320 |

59,400 | Three of a Kind | 103,776 - 288 - 1,320 - 19,008 - 3,960 - 19,800 |

103,776 |
TOTAL |

Almost done. Next, I compute the frequencies for hands containing just a single wild Deuce.

Hands with One Deuce | ||
---|---|---|

48 | Five of a Kind | 4 x 12 |

544 | Straight Flush | 4 x 34 x 4 |

8,448 | Four of a Kind | 4 x C(4,3) x 12 x 44 |

9,504 | Full House | 4 x C(4,2) x C(4,2) x (12x11/2!) |

7,376 | Flush | 4 x C(12,4) x 4 - 544 |

34,272 | Straight | 4 x 34 x 4^{4} - 544 |

253,440 | Three of a Kind | 4 x C(4,2) x 12 x (44x40/2!) |

0 | Two Pair | not possible |

464,688 | One Pair | 4 x C(12,4) x 4^{4} - 544 - 7,376 - 34,272 |

0 | High Card | not possible |

778,320 |
TOTAL |

Lastly, we handle all the cases where a hand has *no* wild cards.

Hands with No Deuces | ||
---|---|---|

0 | Five of a Kind | not possible |

32 | Straight Flush | 8 x 4 |

528 | Four of a Kind | 12 x 44 |

3,168 | Full House | C(4,3) x 12 x C(4,2) x 11 |

3,136 | Flush | 4 x C(12,5) - 32 |

8,160 | Straight | 8 x 4^{5} - 32 |

42,240 | Three of a Kind | 12 x C(4,3) x (44x40/2!) |

95,040 | Two Pair | (12 x C(4,2) x 11 x C(4,2))/2! x 40 |

760,320 | One Pair | 12 x C(4,2) x (44 x 40 x 36)/3! |

799,680 | High Card | (48 x 44 x 40 x 36 x 32)/5! - 32 - 3,136 - 8,160 |

1,712,304 |
TOTAL |

Okay, that was a lot of work, but we have used raw math to enumerate all the various hand frequencies. The final test was to see if the totals obtained from my poker code matched the totals obtained emperically. In other words, if my poker code states that there are 672 Five-of-a-Kind hands, then if I add up all the various ways to make a Five-of-a-Kind using from one to four Deuces, they should be equal.

Breakdown of Five Card Poker Handswith Deuces Wild |
||||||
---|---|---|---|---|---|---|

Hand Rank | No Deuces |
One Deuce |
Two Deuces |
Three Deuces |
Four Deuces |
SUM |

Five of a Kind | 0 | 48 | 288 | 288 | 48 | 672 |

Straight Flush | 32 | 544 | 1,320 | 656 | 0 | 2,552 |

Four of a Kind | 528 | 8,448 | 19,008 | 3,568 | 0 | 31,552 |

Full House | 3,168 | 9,504 | 0 | 0 | 0 | 12,672 |

Flush | 3,136 | 7,376 | 3,960 | 0 | 0 | 14,472 |

Straight | 8,160 | 34,272 | 19,800 | 0 | 0 | 62,232 |

Three of a Kind | 42,240 | 253,440 | 59,400 | 0 | 0 | 355,080 |

Two Pair | 95,040 | 0 | 0 | 0 | 0 | 95,040 |

One Pair | 760,320 | 464,688 | 0 | 0 | 0 | 1,225,008 |

High Card | 799,680 | 0 | 0 | 0 | 0 | 799,680 |

TOTAL |
1,712,304 |
778,320 |
103,776 |
4,512 |
48 |
2,598,960 |

At this point, it might be interesting to see how the "normal" distribution of poker hands changes when we introduce four wild Deuces. As expected, most of the hand count frequencies increase, with the exception of the Two Pair and High Card hands.

Five Card Poker Hands | ||
---|---|---|

Nothing Wild |
Deuces Wild |
Hand Rank |

0 | 672 | Five of a Kind |

40 | 2,552 | Straight Flush |

624 | 31,552 | Four of a Kind |

3,744 | 12,672 | Full House |

5,108 | 14,472 | Flush |

10,200 | 62,232 | Straight |

54,912 | 355,080 | Three of a Kind |

123,552 | 95,040 | Two Pair |

1,098,240 | 1,225,008 | One Pair |

1,302,540 | 799,680 | High Card |

2,598,960 |
2,598,960 |
TOTAL |

One might be tempted to re-order the poker hands based on the
new frequency counts. If you did so, the Four-of-a-Kind hand
would be placed between the Flush and Straight hands, the
Three-of-a-Kind and Two Pair hands would swap places, and so
would the One Pair and High Card hands. However, in every
home game that I've played in that allows wild cards, I have
never seen the hand rankings rearranged based on the "true"
frequencies. In other words, a Flush still loses to a
Four-of-a-Kind, even thought it is actually *harder*
to get a Flush than a Four-of-a-Kind when Deuces are wild.
There are a number of reasons for keeping the poker hand
rankings order unchanged. First, every poker player has
already committed the "correct" ordering of poker hands to memory. Everyone
just *knows* that a Four-of-a-Kind is a powerful hand. Nobody
wants to be forced to remember another ordering just for
wild card games. Plus, some player would inevitably forget
the new ordering, bet his Four-of-a-Kind heavily, and then
get extremely angry when he loses to a Flush or a Full House.
Second, one could always "cheat" by changing the value of
your hand. For example, suppose you hold three wild Deuces
along with the Six and Jack of Spades. Logic would tell
you that you have Four Jacks with a Six kicker. However,
since Full Houses are supposedly harder to get than Four-of-a-Kinds,
you would instead say that you have a Full House, Jacks over
Sixes. It's sort of a Catch-22. If you rearrange the poker
hand orders, then players will just start calling their hands
differently to circumvent the new ordering, which in turn will
screw up the frequencies again, which sort of defeats the
purpose of re-ordering the hands in the first place!
I guess one could get around this trickery (of a hand
having multiple values) by enforcing a "rule" that a player
with wild cards must call his hand as high as possible based
on the "normal" frequencies, even if that means having a
losing hand; but that also would be hard to remember and
cause lots of frustration at showdown. Third, one of the
main reasons of having wild card games is the chance to gleefully
announce that you have a Four-of-a-Kind or a Straight Flush.
Let's face it, that doesn't happen too often in regular poker.
It's great fun to be able to lay down and win with one of
these "power" hands, which wouldn't happen as often if we put the
Four-of-a-Kind in its "proper" place. Lastly, consider the
game of Seven Card Stud (nothing wild). Each player at showdown
will have seven cards, of which they each make the best five
card hand possible. If you calculate the hand frequencies of
all possible C(52,7) combinations, and order them based on their
likelihood of appearing, you get the exact same order as we did
with just five cards, *except* that One Pair is easier to
get than a High Card hand. It's actually easier to get a pair
than seven crappy cards (which explains why the game *Razz* is
so much fun), but the poker community didn't try
to reverse the hand rankings of those two hands when playing
Seven Card Stud. They remained the same. So likewise, even though
wild cards throw the hand rankings all out of whack, the "normal"
ordering is maintained.

Well, I hope you enjoyed this geeky and math-laden journey into the world of poker. I doubt if this information will make you a better poker player, and unless you play in a home game, you probably won't be playing too many games containing wild cards.

*Cactus Kev, copyright 2003*